# Why are some associated Legendre functions not orthogonal to each other?

For example, $$P_1^1 = sin\theta$$ and $$P_2^2 = 3 sin^2\theta$$ seem to be not orthogonal to each other because the integral $$\int_0^\pi (sin\theta)(3 sin^2\theta) sin\theta d\theta$$ is not zero. Am I missing something? Thank you for your help!

• You are expecting an orthogonality relation that doesn’t hold. See en.wikipedia.org/wiki/… for the two relations that do hold. See how the two functions have to have the same $l$ or the same $m$? – G. Smith Nov 20 '19 at 5:45
• normally we treat the associated legendre functions as polynomials that satisfies $\int_{-1}^1P^m_aP^m_b=C(m,a)\delta_{a,b}$ – Ariana Nov 20 '19 at 5:46
• There is no reason to expect them to be orthogonal - they're solutions of different Sturm-Liouville problems, and there is no direct relation between them. – Emilio Pisanty Nov 20 '19 at 6:56

Because the orthogonality is supplied by the azimuthal direction when the two $$z$$-components of the angular momentum differ. This is why we usually bundle them together into the spherical harmonics instead of handling them separately.